superalgebra and (synthetic ) supergeometry
A super-group is the analog in supergeometry of Lie groups in differential geometry.
An affine algebraic super group is the formal dual of a super-commutative Hopf algebra.
A super Lie group is a group object in the category SDiff of supermanifolds, that is a super Lie group.
One useful way to characterize group objects $G$ in the category $SDiff$ of supermanifold is by first sending $G$ with the Yoneda embedding to a presheaf on $SDiff$ and then imposing a lift of $Y(G) : SDiff^{op} \to Set$ through the forgetful functor Grp $\to$ Set that sends a (ordinary) group to its underlying set.
So a group object structure on $G$ is a diagram
This gives for each supermanifold $S$ an ordinary group $(G(S), \cdot)$, so in particular a product operation
Moreover, since morphisms in $Grp$ are group homomorphisms, it follows that for every morphism $f : S \to T$ of supermanifolds we get a commuting diagram
Taken together this means that there is a morphism
of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism $\cdot : G \times G \to G$, which is the product of the group structure on the object $G$ that we are after.
etc.
This way of thinking about supergroups is often explicit in some parts of the literature on supergeometry: some authors define a supergroup or super Lie algebra as a rule that assigns to every Grassmann algebra $A$ over an ordinary vector space an ordinary group $G(A)$ or Lie algebra and to a morphism of Grassmann algebras $A \to B$ covariantly a morphism of groups $G(A) \to G(B)$. But the Grassmann algebra on an $n$-dimensional vector space is naturally isomorphic to the function ring on the supermanifold $\mathbb{R}^{0|n }$. So the definition of supergroups in terms of Grassmann algebras is secretly the same as the above definition in terms of the Yoneda embedding.
also called the super-Heisenberg group
The additive group structure on $\mathbb{R}^{1|1}$ is given on generalized elements in (i.e. in the logic internal to) the topos of sheaves on the category SCartSp? of cartesian superspaces by
Recall how the notation works here: by the Yoneda embedding we have a full and faithful functor
SDiff$\hookrightarrow$ $Fun(SDiff^{op}, Set)$
and we also have the theorem, discussed at supermanifolds, that maps from some $S \in SDiff$ into $\mathbb{R}^{p|q}$ is given by a tuple of $p$ even section $t_i$ and $q$ odd sections $\theta_j$. The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object $S$ it corresponds to under the Yoneda embedding.
Now, for each $S \in$ SDiff there is a group structure on the hom-set $SDiff(S, \mathbb{R}^{1|1}) \simeq C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ given by precisely the above formula for this given $S$
where $(t_i, \theta_i) \in C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ etc and where the addition and product on the right takes place in the function super algebra $C^\infty(S)$.
Since the formula looks the same for all $S$, one often just writes it without mentioning $S$ as above.
The super-translation group is the $(1|1)$-dimensional case of the super Euclidean group.
…
…
There is a finite analog for super-groups that does not quite fit in the framework presented here:
A finite super-group is a tuple $(G, z \in G)$, where $G$ is a finite group and $z$ is central and squares to $1$.
The representations of a finite super-group are $\mathbb{Z}_2$-graded: An irreducible representation has odd degree if $z$ acts by negation, and even degree if it acts as the identity.
This definition is found e.g. in:
Deligne's theorem on tensor categories (see there for details) says that every suitably well-behave linear tensor category is the category of representations of an algebraic supergroup. In particular the Hopf algebra of functions on an affine algebraic supergroup is a triangular Hopf algebra.
Tannaka duality for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-2-module |
sesquialgebra | 2-ring = monoidal presentable category with colimit-preserving tensor product |
bialgebra | strict 2-ring: monoidal category with fiber functor |
Hopf algebra | rigid monoidal category with fiber functor |
hopfish algebra (correct version) | rigid monoidal category (without fiber functor) |
weak Hopf algebra | fusion category with generalized fiber functor |
quasitriangular bialgebra | braided monoidal category with fiber functor |
triangular bialgebra | symmetric monoidal category with fiber functor |
quasitriangular Hopf algebra (quantum group) | rigid braided monoidal category with fiber functor |
triangular Hopf algebra | rigid symmetric monoidal category with fiber functor |
supercommutative Hopf algebra (supergroup) | rigid symmetric monoidal category with fiber functor and Schur smallness |
form Drinfeld double | form Drinfeld center |
trialgebra | Hopf monoidal category |
2-Tannaka duality for module categories over monoidal categories
monoidal category | 2-category of module categories |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-3-module |
Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
monoidal 2-category | 3-category of module 2-categories |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
Discussion via functorial geometry:
See also
Katsumi Yagi, Super Lie Groups, Adv. Stud. Pure Math. Progress in Differential Geometry (1993) 407 - 412 (euclid:1534359537)
Dennis Westra, Superrings and supergroups, 2009 (pdf)
Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)
Veeravalli Varadarajan, section 6.1 of: Supersymmetry for mathematicians: An introduction
Discussion of group extensions of supergroups includes
Discussion as Hopf-superalgebras includes